Yes, they're on Genosha. I couldn't find a way to shoehorn it into the fic without resorting to an infodump, but in my head, Erik's Brotherhood is at a point where they're beginning to transition from a militant organisation into a government. Insofar that Erik is ruling Genosha, it's really more out of a lack of any other contender than anything else -- the mutants had pushed out the non-mutant inhabitants, but they haven't actually gotten around to creating an actual nation-state.

My take on the alpha-omega dynamics in this fic was extrapolated from the prompt: law or no law, the tradition of alphas "owning" omegas is pretty much ingrained culturally, to a lesser or greater extent depending on where you are. Erik could've laid down a rule fully emancipating mutant omegas on Genosha, but... he didn't. He never even thought about it. Hence Charles being gently mocking/bitter about Erik's ideology: hurray for mutant supremacy or whatever, but from where he's standing, there's not much difference between "humans" and mutants -- genetically and culturally. Certainly not where omegas are concerned.

You've nailed it re: Mystique. I like to think that when the chips are down, she'll side with Charles, but she hasn't been pushed to that point just yet. Right now, to her it's all about Charles and what he specifically went through, but she doesn't see that if the way this world works isn't so stacked against omegas, Charles might never be so damaged in the first place.